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We obtain a one-parameter family of (q, p)-representations of quantum mechanics; the Wigner distribution function and the distribution function we previously derived are particular cases in this family. We find the solutions of the evolution equations for the microscopic classical and quantum distribution functions in the form of integrals over paths in a phase space. We show that when varying canonical variables in the Green's function of the quantum Liouville equation, we must use the total increment of the action functional in its path-integral representation, whereas in the Green's function of the classical Liouville equation, the linear part of the increment is sufficient. A correspondence between the classical and quantum schemes holds only under a certain choice of the value of the distribution family parameter. This value corresponds to the distribution function previously found.

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